The Cooper pairing mechanism of heavy fermionsuperconductors1,2,3,4, long thought to be due to spin fluctuations5,6,7, has not yet been determined. It is the momentum space (k-space) structure of the superconducting energy gap Δ(k) that encodes specifics of this pairing mechanism. However, because the energy scales are so low, it has not been possible to directly measure Δ(k) for any heavy fermion superconductor. Bogoliubov quasiparticle interference imaging8, a proven technique for measuring the energy gaps of superconductors with high critical temperatures9,10,11, has recently been proposed12 as a new method to measure Δ(k) in heavy fermion superconductors, specifically CeCoIn5 (ref. 13). By implementing this method, we detect a superconducting energy gap whose nodes are oriented along k∥(±1,±1)π/a0 directions14,15,16,17. Moreover, for the first time in any heavy fermion superconductor, we determine the detailed structure of its multiband energy gaps Δi(k). For CeCoIn5, this information includes: the complex band structure and Fermi surface of the hybridized heavy bands, the fact that largest magnitude Δ(k) opens on a high- k band so that the primary gap nodes occur at unforeseen k-space locations, and that the Bogoliubov quasiparticle interference patterns are most consistent with gap symmetry. Such quantitative knowledge of both the heavy band-structure and superconducting gap-structure will be critical in identifying the microscopic pairing mechanism of heavy fermion superconductivity.
The heavy fermion superconductor CeCoIn5 (ref. 13) has a crystal unit cell with a = b = 4.6 Å, c = 7.51 Å, as shown schematically in Fig. 1a, and a superconducting critical temperature Tc = 2.3 K. If antiferromagnetically ordered, the Ce3+ atoms would exhibit local magnetic moments μ = 0.15 μB (ref. 18). Although that state does not exist in the pure compound studied here, antiferromagnetic spin fluctuations do persist19. Pioneering research on CeCoIn5, using the recently introduced heavy fermion interference imaging technique20,21, reveals a heavy band22 at T≈20 K, in qualitative agreement with angle resolved photoemission measurements at similar temperatures23,24. However, no detailed and quantitative determinations of the heavy fermion band structures or Fermi surfaces have been reported for this compound. In its superconducting phase, the Cooper pairs are spin singlets25,26, so that an even parity Δ(k) is required. The magnetic field-angle dependence of thermal conductivity15 and specific heat17 are interpreted as evidence of energy-gap nodes |Δ(k)| = 0for momentum space directions k∥(±1,±1)π/a0. A fully detailed knowledge of the k-space structure of Δ(k) is required to understand the microscopic Cooper pairing mechanism of heavy fermions. This cannot be achieved using such indirect methods, or by using photoemission because the energy resolution required is δ E<100 μeV. Motivated thus, high-resolution Bogoliubov quasiparticle scattering interference imaging has recently been mooted12 as a promising approach for determining Δ(k) of heavy fermion superconductors, specifically for CeCoIn5.
There are three elements of k-space electronic structure expected in a generic heavy fermion superconductor4. First, the high-temperature state consists of a conventional (light) electronic band, indicated schematically by the dashed curve in Fig. 1b, that coexists with localized f-electron states on each magnetic atom. At lower temperatures, hybridization between this light band and the f-electron states results in its splitting into two new heavy bands, as shown schematically by the solid blue lines in Fig. 1b. The right panel shows how the resulting very flat bands generate a greatly enhanced density-of-electronic-states N(E) within a few meV of EF—hence the ‘heavy’ effects seen in thermodynamic studies. At least one of these heavy bands crosses EF at the new Fermi wave vector kFH, as shown within the green box in Fig. 1b. It is in this region of k-space that, at even lower temperatures, the heavy quasiparticles are hypothesized to bind into heavy Cooper pairs. An energetically particle–hole symmetric superconducting energy gap Δ(k), probably of an unconventional nature1,2,3,4,5,6,7, is then expected to open in the heavy quasiparticle spectrum at the Fermi surface, as shown schematically in Fig. 1c. The right panel shows the further expected changes in N(E) for a nodal Δ(k).
To search for this sequence of phenomena, we use pure CeCoIn5 samples inserted into the cryogenic ultrahigh vacuum of a 3He-refrigerator-based spectroscopic imaging scanning tunnelling microscope27 (SI-STM), and mechanically cleaved therein. Atomically flat a–b surfaces are achieved; a typical resulting topograph with visible atomic lattice a0 = 4.6 Å (that was previously assigned to be the Ce lattice22) is shown in Fig. 1d. On all such surfaces, the density-of-states is determined from the spatially averaged differential tunnelling conductance measured far from impurity atoms. Whereas the basic of the unhybridized ‘light’ bands is measured over the range |E|≤200 meV (Fig. 1e), the complex scattering interference features associated with the heavy band structure are only visible within the range ∼−4 meV<E<12 meV. Vertical arrows in Fig. 1f then indicate the limits of the hybridization gap Eh for CeCoIn5, as determined directly from the heavy band scattering interference analysis in Fig. 3.
On entering the superconducting phase, develops an energy gap with a maximum value |Δmax| = 600±50 μeV, a V-shaped that is the signature of a nodal28,29 Δ(k), and a finite28,29,30 N(E = 0), all as shown in Fig. 1g. Figure 2a shows a typical example of atomically resolved images g(r,E)≡dI/dV (r,E = eV) measured within the superconducting gap at E = 250 μeV, and acquired in a 32 nm×32 nm field of view (FOV). The superconducting peak-to-peak gap map Δpp(r) in the same FOV (Fig. 2b) reveals the electronic homogeneity of this material. In Fig. 2c we show an image of the CeCoIn5 Abrikosov vortex array acquired at T = 250 mK, B = 3 T in a larger FOV; its shape and orientation are in excellent agreement with small-angle neutron scattering studies31. As all these phenomena disappear at the superconducting Tc observed in bulk measurements, the |Δpp| = 550±50 μeV energy gap with V-shaped N(E) is definitely that of the superconductor. Determination of the k-space structure of Δ(k) for CeCoIn5 is then the main focus of this paper.
To proceed, we image the differential conductance g(r,E) with atomic resolution and register, and then determine g(q,E), the square root of the power spectral density Fourier transform of each image. The measurements are all carried out at 250 mK—thereby achieving an energy resolution δ E∼3.5 kBT∼75 μeV. By using a >30 nm-square FOV we simultaneously achieve q-space resolution |δ q|<2% of the reciprocal unit cell, and a concomitant k-space resolution |δ k|<2% of the Brillouin zone. To investigate both how the light bands transform to hybridized heavy fermion states, and how superconductivity then emerges, we measure these datasets on three distinct energy scales, each of about an order of magnitude smaller energy range than the previous one, as described in the Supplementary Information. These data are used to evaluate elements of k-space electronic structure, based on the fact that elastic scattering of electrons with momentum −k(E) to +k(E) generates interference patterns occurring as maxima at q(E) = 2k(E) in g(q,E), an effect recently revealed to exist even when hybridization generates heavy fermion bands20,21,22. In Supplementary Section SI and Fig. S1 we show the measured g(q,E) at T = 1.2 K, for −100 meV<E<30 meV, focusing on the light unhybridized electronic structure. Here the maximum intensity features move slowly and smoothly to smaller |q|-radii with increasing E, thereby revealing a light and simple tetragonal band (Supplementary Section SI and Movie S1).
Significant departures from this simple phenomenology are found to occur only within the energy range ∼−4 meV<E<12 meV. In Fig. 3a–e we next show the measured g(q,E) at T = 250 mK within this range (Supplementary Section SI). The onset of hybridization is detected as a sudden transformation of the previously unchanging structure of g(q,E) occurring at E≈−4 meV (Fig. 3b), followed by a rapid evolution of the maximum intensity features (indicated by circles and arrows Fig. 3c) towards smaller |q|-radius interference patterns. Then, in Fig. 3c we see that an abrupt jump to a larger |q|-radius occurs, followed by a second rapid diminution of interference pattern |q|-radii in Fig. 3c–e (complete phenomena shown in Supplementary Section S1 and Movie S1). These are all the expected quasiparticle interference (QPI) signatures of the appearance of hybridized heavy fermion bands. Thus, for CeCoIn5 this approach reveals how the light conduction band is split into two heavy bands within the hybridization gap −4 meV<Eh<12 meV. To see this directly, we show in Fig. 3f,g the measured evolution of the maxima in g(q,E) for two directions in q-space. The light band (grey dots) begins to deviate near −4 meV towards the lower heavy band which crosses EF at smaller |q| = 2|kFH|, and evolves quickly to even smaller |q|(blue dots). Within a few meV above EF, the interference patterns jump to a much larger |q| and then evolve (blue dots) back towards the light band (grey dots), which they rejoin near +12 meV. This heavy band actually crosses below E = 0 at high k, producing an electron-like Fermi surface whose intra-band scattering interference generates interference patterns at low q (blue dots E<0 as |q|→0). These data (Fig. 3a–e, Supplementary Section SI), and the extracted dispersions (Fig. 3f,g) are next used to determine details of the heavy fermion band structure.
In general for a complex and multi-band k-space structure, achieving a deterministic inversion procedure from g(q,E) data to the complete band structure can be challenging11. Here, comparison of the predicted scattering interference dispersions |q(E)| from a specific model of the heavy bands described in Supplementary Section SII and Fig. 3h, with the experimental |q(E)| data within the hybridization range Eh, reveals good agreement (Supplementary Figs S2 and S3). The critical elements in our model that lead to this agreement are the nearly parallel sections of the light band contours-of-constant-energy and the hybridization with a specifically shaped f-band. Some of these same elements of k-space structure can equally be found in ref. 12. Our model concentrates on best emulation of the key empirical phenomena of the heavy QPI data. On this basis, the g(q,E) in Fig. 3a–g are used to motivate the detailed k-space model for the heavy bands of CeCoIn5, as shown in Fig. 3h (Supplementary Section SII). Here, within the rangeEh, a light-hole-like band centred around Γ (or equivalently M) hybridizes with a localized f-electron band (Supplementary Section SII). The resulting lower heavy band β has a simple Fermi surface and closes quickly above EF, whereas the upper heavy band α is highly anisotropic with a complex Fermi surface as it crosses below EF, making it effectively electron like. Our model indicates the possibility of a small dimple that crosses back above EF, but this is in no way critical to the subsequent analysis.The Fermi surfaces are shown as solid lines on the E = 0 planes of Fig. 3h. Their relationship with the FS areas deduced from quantum oscillations (QO) at high fields32,33 is discussed in Supplementary Section VI.
To investigate the superconductivity on the heavy bands in Fig. 3h, Fig. 4a–e shows the measured g(q,E) at T = 250 mK and |E|<300 μeV, within the superconducting energy gap. Here we see extremely rapid evolution in g(q,E) over energies of a few 100 μeV, and the appearance of a four-fold symmetric ‘nodal’ g(q,E) structure as E→0. Clearly, this g(q,E = 0) exhibits far more complexity than expected for a single-band nodal superconducting energy gap. To explore these phenomena we carry out Bogoliubov QPI simulations based on the two heavy bands, α and β (Figs 3h,4k), but now specifying their superconducting energy gaps Δα(k) and Δβ(k), whose derivation is discussed below. Here the inter-nodal scattering wave vectors for the α band (coloured arrows Fig. 4k) are demonstrably consistent with the measured inter-nodal scattering vectors in g(q,E = 0) data, whereas the equivalent internodal signatures are undetected for the β band. As specific heat data show that all the main bands in CeCoIn5 are gapped at lowest temperatures13, this suggests that the gap on the β band, although extant, is too small to be detected at T∼250 mK (ref. 34) because the energy uncertainty of tunnelling electrons δ E∼75 μeV probably exceeds the gap maximum. This situation also provides a simple and plausible explanation for the non-zero tunnelling conductance at E = 0. What our data do indicate is that the primary gap of CeCoIn5 actually occurs on the high- k α-band with lines of gap-nodes along the k = (0,0)→(±π,±π)/a0 directions, so that the dominant gap nodes in CeCoIn5 occur at unforeseen k-space locations (Fig. 4k).
Next we consider a detailed comparison of the measured g(q,E) data for |E|≤600 μeV at T∼250 mK with theoretical simulations of Bogoliubov QPI in g(q,E) using the α,β Fermi surfaces described in Figs 3h,4k. The simulations have been carried out using various symmetries for the superconducting energy gaps and are described in full detail in Supplementary Section SII. Our model with superconducting gaps of symmetry given approximately by |Δβ(θk)|<50 μeV and Δα(θk) = Acos(2θk), with A = 600±50 μeV, yields a set of simulated g(q,E) that are far more consistent with the experimental data than any of the other models we have considered (Supplementary Fig. S5, Supplementary Section SII). For comparison, a direct experimental estimation of |Δ(θk)| can be achieved by using g(q,E) data in a procedure largely independent of the Fermi surface details. To obtain the angle θq dependence of the energy gap that opens at Tc, we integrate the total spectral weight g(q,E) within a given |δ q| range containing the Fermi surface, with the lowest |q| large enough to exclude effects of heterogeneity and the largest |q| small enough to exclude the Bragg peaks. A clear gap Δ(θq) is observed to open in this integral of g(q,E) on passing below Tc, as demonstrated in Supplementary Section SIII. In Fig. 4l we plot the measured energy gap |Δ(θq)| from this technique (red dots) along with Δα(θk) = |Acos(2θk)| with A = 600 μeV (solid line). Their agreement provides strong independent motivation for our gap structure model (Fig. 4k and Supplementary Section SII). A final stimulating observation is that the departures in the Δ(θq) at a higher energy from the simple Δα(θk) = Acos(2θk) might be expected if high q scattering between these locations on the α band is involved in the Cooper pairing mechanism.
Overall, these data represent the first measurements of the k-space structure of the superconducting energy gaps Δi(k) for any heavy fermion superconductor. They reveal a wealth of previously unknown information on the Δi(k) of CeCoIn5, including: the complex Fermi surface of the hybridized heavy bands (Figs 3h,4k); the spectroscopic signature of four nodal lines in |Δ(k)| oriented along k = (±1,±1)π/a0 or Ce–Ce directions14,15,16,17; that the dominant Δ(k) opens on the α heavy band at high k (Fig. 4k); the unforeseen k-space locations of the dominant gap nodes that ensues (Fig. 4k); that the Bogoliubov QPI patterns are most consistent with gap symmetry; and evidence for a departure in Δ(k) from a simple cos(2θk) dependence at regions of the α band connected by q≈(±1,±1)π/a0 (Fig. 4k,l). These detailed multi-band Δi(k) data provide information critical for determining the microscopic mechanism of unconventional superconductivity in CeCoIn5. Moreover, the novel approach proposed in ref. 12, and established here, reveals an exciting new opportunity to achieve a quantitative understanding of the microscopic physics of heavy fermion superconductivity in general.
High-quality CeCoIn5 single crystals were grown at BNL (details in ref. 13). Magnetization measurements before sample insertion into the STM show a sharp transition with Tc = 2.1 K. The samples were mechanically cleaved in cryogenic ultrahigh vacuum at T∼10 K and directly inserted into the STM head at 4.2 K. Etched atomically sharp and stable tungsten tips with an energy-independent density of states are used. Differential conductance measurements throughout used a standard lock-in amplifier. See Supplementary Information for additional details on data treatment and extraction.
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We are particularly grateful to I. Eremin, J. E. Hoffman, D-H. Lee and A. R. Schmidt for advice and discussions. We acknowledge and thank A. Akbari, M. Aprili, M. H. Fischer, M. Hamidian, E-A. Kim, S. A. Kivelson, M. Norman, J. P. Reid, D-H. Lee, D. J. Scalapino and K. Shen for helpful discussions, advice and communications. Supported by US DOE under contract number DEAC02-98CH10886 (J.C.D. and C.P.) and under Award No. DE-FG02-05ER46225 (D.K.M., J.v.D.); by the UK EPSRC under programme grant ‘Topological Protection and Non-equilibrium States in Correlated Electron Systems’ (A.R., A.P.M.); M.P.A. acknowledges support through the ETH Fellowship program; A.P.M. acknowledges support of a Royal Society-Wolfson Award.
The authors declare no competing financial interests.
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Allan, M., Massee, F., Morr, D. et al. Imaging Cooper pairing of heavy fermions in CeCoIn5. Nature Phys 9, 468–473 (2013). https://doi.org/10.1038/nphys2671
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